A positive definite kernel \(\kappa \)  is said to be perfect if the energy and consistency principles are both fulfilled [1]}. It follows that for a perfect kernel \(\kappa \) , the cone \(\mathcal {E}^+\)  is complete in the induced strong topology, and moreover every strong Cauchy net \((\mu _s)\subset \mathcal {E}^+\)  converges to the same unique limit \(\mu _0\in \mathcal {E}^+\)  both strongly and vaguely. In fact, all the vague limit points of such \((\mu _s)\)  must coincide, being equal to the (unique) strong limit \(\mu _0\) ; since the vague topology on \(\mathfrak {M}\)  is Hausdorff, the same \(\mu _0\)  must also serve as the vague limit of \((\mu _s)\)  (see [2]}).
